Collaborative Research: A Posteriori Error Analysis for Complex Models with Applications to Efficient Numerical Solution and Uncertainty Quantification

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Many scientific and engineering problems of importance to the nation's infrastructure and defense are concerned with multi-physics systems in which multiple physical processes interact in complex ways. An important example is the flow of a liquid transporting reacting chemicals in which the reaction affects the fluid properties of the liquid. Such reacting flows arise in applications ranging from biological systems to combustion processes associated with energy use. In general, the complexity of multi-physics systems prevents direct experimental observation of crucial features. Thus their study depends critically on computing approximate solutions of mathematical models describing the processes and their interactions. However, such simulations strain the computational capabilities of the most powerful computers and consequently, computational errors in the approximations are always significant and may be overwhelming. Over two decades, the project investigators have developed a systematic approach for producing accurate computational estimates of the error of approximate solutions of models of multi-physics systems. In this project, the investigators explore the use of error estimates from this approach to guide the efficient use of computational resources in order to maximize the fidelity of approximate solutions of multi-physics systems. They also apply the error estimates to accurately quantify the uncertainty in predictions of behavior of multi-physics systems based on the approximate solutions of models. The results of this project will enhance the ability of the nation's engineers and scientists to investigate and predict the behavior of complex physical systems important to the nation's security and infrastructure. This project tackles critical problems associated with using sophisticated cutting-edge multi-discretization numerical methods for multiscale, multiphysics models to pursue scientific inference and engineering design. The research is based on a posteriori error analysis for multi-physics, multi-discretization problems that quantifies the effects of a wide variety of discretization steps through the use of adjoint problems and computable residuals. The primary focus of the project is twofold: (1) Developing and analyzing methods for using accurate error estimates to guide discretization choices in order to achieve a desired accuracy at roughly minimal computational cost; and (2) Investigating how to extend accurate error estimation methods to address uncertainty quantification for multiphysics systems, where 'discretization' includes the sampling of a random process and the overall error is a combination of discretization and sampling errors. The investigators pursue the development of novel multi-stage approaches to the construction of efficient numerical solutions and the extension of a posteriori error analysis to statistical computations. The project also involves the extension of the theory of a posteriori error analysis to hyperbolic problems and nonstandard quantities of interest.